Abstract

We propose a scheme of photon blockade in a system comprising of coupled cavities embedded in Kerr nonlinear material, where two cavities are driven and dissipated. We analytically derive the exact optimal conditions for strong photon antibunching, which are in good agreement with those obtained by numerical simulations. We find that conventional and unconventional photon blockades have controllable flexibilities by tuning the strength ratio and relative phase between two complex driving fields. Such unconventional photon-blockade effects are ascribed to the quantum interference effect to avoid two-photon excitation of the coupled cavities. We also discuss the statistical properties of the photons under given optimal conditions. Our results provide a promising platform for the coherent manipulation of photon blockade, which has potential applications for quantum information processing and quantum optical devices.

© 2015 Optical Society of America

1. Introduction

The generation of nonclassical states of photons [1, 2] represents a hot research topic in integrated quantum technologies and quantum optics and is vital for furture applications. In particular, the single-photon source plays an important role in quantum information technologies, and makes their production becoming an ultimate goal of defining a photonic-based architecture in quantum communication [3], quantum metrology [4], and quantum information technologies [5, 6]. The physics behind this mechanism is photon blockade effect, which has been demonstrated in an optical cavity coupled to a quantum emitter [7–10] and coupled single quantum dot-cavity system [11]. Subsequently, a sequence of experimental groups observed the strong antibunching behaviors in different systems, such as a quantum dot in a photonic crystal [12], circuit cavity quantum electrodynamics systems [13, 14].

It is well known that a coherent light beam enters a nonlinear system whose effective nonlinear response is able to produce a shift of the two-photon resonance that is larger than its linewidth [15], which is named as conventional photon blockade. Typical examples include quantum optomechanical systems [16–26] and cavity quantum electrodynamics [27–36]. A sequence of schemes have been proposed in nanostructured cavities and semiconductor microcavities with second-order nonlinear [37, 38]. Such photon blockade has been applied to transistors [39], interferometers [40], and optical rectifiers [41, 42], to mention a few.

The unconventional photon blockade effect was first predicted [43] in photon molecule system [44, 45], which shows that the presence of two photons in two coupled cavities with weak nonlinearities is prevented by destructive quantum interference between different paths in the nonlinear coupled cavity system [46–52]. Similar effects recently were considered in various models, such as coupled single-mode cavities with second- or third-order nonlinearity [53–58], double asymmetric cavities [59], Gaussian squeezed states [60], optical cavity with a quantum dot [61, 62], coupled optomechanical systems [63, 64], and bimodal coupled polaritonic cavities [65, 66], and so on.

Here, we show that an unconventional photon blockade can be engineered in two coupled cavities filled with nonlinear Kerr mediums, where two cavities are driven and dissipated. The approximate optimal conditions for the detuning Δ and Kerr nonlinearity U have been derived when Jκ and φ = 0 in [47]. Here we consider a more general scenario, in which we remove these two constraints, and derive the exact optimal conditions in order to generate strong antibunching photons in the system for all the regime of parameters. There are a total of eight sets of solutions for the optimal detuning since the discriminant changes its sign as the strength ratio and the relative phase vary between two complex driving fields, which can be controlled precisely [67]. Hence exact optimal conditions give a fully description of photon blockade, and holds for all range of parameters. On the contrary, if there are no exact optimal conditions, it is very difficult to experimentally find the all eight optimal regimes of photon blockade. Our results provide a promising platform for the coherent manipulation of photon blockade, which has potential applications for quantum information processing and quantum optical devices.

2. Physical model and optimal antibunching conditions

The system under consideration is composed of two tunnel-coupled cavities filled with a Kerrtype nonlinear material. We assume that each cavity to be single mode, described by Bose operator â and b^, respectively. The nonlinear second-quantized Hamiltonian reads [43, 46, 68]

H^0=h¯ωaa^a^+h¯ωbb^b^+h¯J(a^b^+b^a^)+Uaa^a^a^a^+Ubb^b^b^b^,
where J denotes the tunnel coupling rate, ωj and Uj eigenfrequency and Kerr nonlinearity strength for cavity j (j = a,b), respectively. The setup is illustrated in Fig. 1(a), where the driving frequency is denoted by ωL, and the complex driving strength denoted by Fjeiφj. The total Hamiltonian including the driving terms is then,
H^dr(t)=H^0+(h¯Faa^eiφaiωLt+h¯Fbb^eiφbiωLt+H.c.),
which can be rotated with respect to the external laser frequency ωL
H^=h¯Δaa^a^+h¯Δbb^b^+h¯J(a^b^+a^b^)+Uaa^a^a^a^+Ubb^b^b^b^+(h¯Faeiφaa^+h¯Fbeiφbb^+H.c.),
where Δa = ωaωL and Δb = ωbωL denote the detunings of cavity a and b from two driving fields, respectively. Here we note that the relative phase φ = φaφb plays an importance role, which can be found by a^a^eiφb b^b^eiφb. This also can be seen from the derived optimal conditions (21)(27). Considering the dissipations to the system, the density matrix ρ of the system is governed by the master equation,
ρ˙=i[H^,ρ]+κaD(a^)ρ+κbD(b^)ρ,
where Ĥ is given by Eq. (3), κa and κb denote loss rates for the two cavities, respectively. The superoperator is defined by D(o^)ρ=o^ρo^12o^o^ρ12ρo^o^. The equal-time second-order correlation function for mode a can be given by [69, 70],
ga(2)(0)=a^2a^2a^a^2.

 

Fig. 1 (a) Scheme diagram of the setup. Two cavities are driven by external fields with the complex Rabi frequency Faeiφa and Fbeiφb, respectively. (b) The equal-time second-order correlation function ga(2)(0) is plotted by numerically solving the master Eq. (4). The nearly perfect antibunching can be obtained at U = 1.593 × 10−3κ with φ = 1.6 rad. The other parameters are chosen as Δa = Δb = 0.1539κ, J = 7κ, Ua = 0, and Fa = 0.7κ, Fb = 0.028κ, φ = 1.6rad for blue-bold line, φ = 1.8rad for dashed-green line, φ = 1.4rad for red-dashed-dotted line. (c) The first (second) number in |mn〉 denotes the photon number in cavity a (cavity b). The transition paths (two dashed lines with arrows) lead to the quantum interference (indicated by the ellipse) for the strong photon antibunching.

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We present the variation of ga(2)(0) defined by Eq. (5) as a function of U and Δ by numerically solving master equation (4) in Figs. 1 and 2. First we discuss the features of Fig. 1(b). We find there are three interesting points at U = 1.593 × 10−3κ, 2.03 × 10−3κ, and 1.18 10−3κ marked respectively by A, B,C. Points A, B, and C correspond to different relative phases between two complex driving fields, which are denoted by φ = 1.6 rad, 1.8 rad, and 1.4 rad, respectively.

 

Fig. 2 Dependence of logarithmic plot for ga(2)(0) on U (UUb) and Δ by numerically solving master Eq. (4). Defining FbrFa. Parameters chosen are J = 7κ, Fa = 0.7κ, Ua = 0, and r = 0.02, φ = 2rad for (a), and r = 0.2, φ = 1rad for(b)–(d). We have an interesting observation that there are two and four minimum values for two-order correlation function for (a) [see points P1 and P2 denoted by blue-star] and (b)–(d) [see points P3-P6 denoted by blue-star], respectively. Therefore points P1-P4 correspond to unconventional single-photon blockade, which is smaller than the mode broadening κ, while points P5 and P6 indicate conventional single-photon blockade, which is larger than the mode broadening κ.

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Interestingly, we find that there are two minimum values (see points P1-P) for ga(2)(0) in Fig. 2(a) for given strength ratio r = Fb/Fa, decay rate κa = κbκ, and the relative phase φ. However, ga(2)(0) appears four minimum values (see points P3-P6) in Figs. 2(b)–2(d) as r and φ vary. Therein points P1-P4 correspond to unconventional single-photon blockade, where Kerr nonlinearity U is smaller than the mode broadening κ, while points P5 and P6 indicate conventional single-photon blockade, where Kerr nonlinearity U is larger than the mode broadening κ. One may wonder why there are four points in Figs. 2(b)–2(d) appearing for single-photon blockade under the same parameters? In the following, we will answer this question.

Assume that the system is initially prepared in |00〉, and only these levels are occupied due to two weak complex driving fields. Therefore the total sate of the system can be written as [46, 71],

|Ψ=A00|00+A10|10+A01|01+A20|20+A02|02+A11|11,
describes the dynamics by evolving |Ψ〉 under the action of the non-Hermitian Hamiltonian H˜=H˜i[κaa^a^+κbb^b^]/2. Substituting Eq. (6) into the Schrödinger equation it|Ψ=H˜|Ψ ( h¯=1, hereafter), we arrive at,
itA00=FaeiφaA10+FbeiφbA01,
itA10=δaA10+JA01+FaeiφaA00+AbeiφbA11+2FaeiφaA20,
itA01=δbA01+JA10+FbeiφbA00+FaeiφaA11+2FbeiφbA02,
itA11=(δa+δb)A11+2J(A20+A02)+FbeiφbA10+FaeiφaA01,
itA20=2(δa+Ua)A20+2JA11+2FaeiφaA10,
itA02=2(δb+Ub)A02+2JA11+2FbeiφbA01,
where δj = Δjj/2. Under the weak driving condition, we have |A00| ≫ |A10|, |A01| ≫ |A20|, |A11|, |A02|. In steady state, tAjk=0, the relations between A10 and A01 read,
δaA10+JA01=FaeiφaA00,JA10+δbA01=FbeiφbA00,
and two-photon probability amplitude reduces to
0=2(δa+Ua)A20+2JA11+2FaeiφaA10,0=2(δb+Ub)A02+2JA11+2FbeiφbA01,0=(δa+δb)A11+2J(A20+A02)+FbeiφbA10+FaeiφaA01.

From Eq. (13)A10 and A01 can be written as,

A01A10=eiφaFaJeiφbFbδaeiφbFbJeiφaFaδbζ.

The equal-time second-order correlation function ga(2)(0) can be given by substituting Eq. (6) into Eq. (5) in weak driving limit.

ga(2)(0)2|A20|2|A10|4.

The conditions for ga(2)(0)1 are derived from Eq. (14) by setting A20 = 0. Hereafter, we will set UbU, Ua = 0 because Ua does not appear in the coupled Eq. (14) after taking A20 = 0.

For simplicity, in the following we mainly discuss the case of Δa = Δb ≡ Δ, κa = κbκ, Fb/Fa = r, φaφb = φ. Complementally, in Appendix A we discuss the general situation of Δa ≠ Δb and κaκb, and give the optimal conditions for strong photon antibunching.

Noticing Eq. (15), the conditions for ga(2)(0)0 are derived from Eq. (14) by setting A20 = 0, so we arrive at,

0=2JA11+2FaeiφaA10,0=2(δb+Ub)A02+2JA11+2FbeiφbζA10,0=(δa+δb)A11+2JA02+FbeiφbA10+FaeiφaζA10.

The condition for A02, A11, and A10 to have nontrivial solutions is that the determinant of the coefficient matrix of Eq. (17) is equal to zero, then we obtain

0=4ir2J2[κ+i(U+2Δ)]+e2iφ[(κ+2iΔ)2(2U+2Δiκ)4J2U]4reiφJ(κ+2iΔ)[κ+2i(U+Δ)].

After a simple calculation, we obtain the equation for the detuning Δ:

Δ4+4bΔ3+6cΔ2+4dΔ+e=0,
where these coefficients take b = rJ cos(φ)[rJ sin(φ) − 2κ]/2κ, c=J212(3+8r2)+κ212+rJ12κ[rJκcos(2φ)4(J2+r2J2+κ2)sin(φ)], d = rJ cos(φ)[rJ(4J2 + 5κ2) sin(φ) − 4(1 + r2)J2κ − 2κ3]/8κ, e=12r4J4+18(8r21)J2κ2+116κ4+18rJ[rJ(4J23κ2)cos(2φ)4κ(3J2r2J2+κ2)sin(φ)]. The nature of roots for Eq. (19) is mainly determined by its discriminant. When the discriminant
δ=m327n2<0
with m = e−4bd+3c2 and n = (4h3hmg2), Eq. (19) under this condition has two real roots,
Δopt(1),(2)=[bsgn(g)]λ±|g|/λλ+3h,
then the Kerr nonlinearity for optimal conditions is,
Uopt=4rJv2+2Δ(3κ24Δ2)sin(2φ)+v34cos(φ)v18Δ[κcos(2φ)+2rJsin(φ)],
where ΔΔopt(j), j = 1,2 in Eq. (21). Other parameters take v1 = 2rJκ + (2J2κ2 + 4Δ2)sin(φ),v2 = rJκκ2 sin(φ) + 4Δ2 sin(φ),v3 = −16rJκΔcos(φ) − κ(κ2 − 12Δ2)cos(2φ), h = b2c,.g = d – 3bc + 2b3, λ=[n+δ/273+nδ/273]/2+h. Set η = 12h2m.

When

δ=m327n2>0,
if and only if
h>0,η>0,
are simultaneously satisfied, Eq. (19) has four real roots,
Δopt(3),(4)=b+sγ1±γ2±γ3,
Δopt(5),(6)=bsγ1±γ2γ3.
otherwise, Eq. (19) has four conjugate complex roots.

With Eqs. (25) and (26), the optimal Kerr nonlinearity is expressed as,

Uopt=4rJv2+2Δ(3κ24Δ2)sin(2φ)+v34cos(φ)v18Δ[κcos(2φ)+2rJsin(φ)],
where ΔΔopt(k), k = 3,4,5,6 in Eqs. (25) and (26). Other parameters take v1 = 2rJκ + (2J2κ2 + 4Δ2)sin(φ), v2 = rJκκ2 sin(φ) + 4Δ2 sin(φ),v3 = −16rJκΔcos(φ) − κ(κ2 − 12Δ2)cos(2φ), β=arccos[n/|m|3/27], y1=|m|/3cos(β/3)+h, y2,3=|m|/3cos(β/3±2π/3)+h. Here s = −sgn(g) when all yj are positive, otherwise s = sgn(g).

Here we address that when the optimal conditions (21) and (22) [or (25), (26) and (27)] are simultaneously satisfied, strong antibunching can be obtained, otherwise the system is not in strong antibunching regimes.

Now we begin to discuss the features of Figs. 1(b) and 2 applying the optimal conditions Eqs. (21), (25) and (26). In Fig. 1(b), the discriminant δ is less than zero, Δ = 0.1539κ or −0.1664κ by Eq. (21) when φ = 1.6 rad, which explains the point A in Fig. 1. Δ = −0.2134κ or 0.1161κ at φ = 1.8κ rad explains point B when δ < 0. The explanation for the point C is the same as point B.

In Fig. 2, we plot ga(2)(0) as a function of U and Δ by numerically solving the master Eq. (4). We find that these optimal points P1-P6 are in good agreement with those obtained by the effective Hamiltonian method. For unconventional single-photon blockade regimes, see Eqs. (21) and (22) [i.e., for (a), point P1:(Δopt(2),Uopt)=(0.276κ,0.00436κ), point P2:(Δopt(1),Uopt)=(0.1945κ,0.004453κ) when δ < 0], Eqs. (25)(27) [i.e., in (b), point P3:(Δopt(6),Uopt)=(0.1614κ,0.02705κ), point P4:(Δopt(4),Uopt)=(0.4439κ,0.01518κ) when δ > 0]. For the conventional single-photon blockade, Δopt and Δopt take (Δopt(3),Uopt)=(6.9165κ,4.222κ) (denoted by point P5), and (Δopt(5),Uopt)(6.81κ,4.78κ) (denoted by point P6), which are shown in the Figs. 2(c) and 2(d), respectively. Although the optimal values can be found by numerically solving the master equation, the analytical optimal values given by the effective Hamiltonian have the following advantages. Firstly, the analytical optimal U and Δ (21)(27) are convenient to analyze the regime of antibunching, this would help in experiments to find the required parameters. Secondly, by the analytical expressions for Uopt and Δopt, we can find the roles the system parameters playing. Hence exact optimal conditions (21)(27) give a fully detailed description of photon blockade, and hold for all range of parameters. On the contrary, without these exact optimal conditions, it is very difficult to experimentally find all of the optimal regimes of photon blockade.

Physically, in order to understand the conventional and unconventional single-photon blockade, we sketch the corresponding bare energy levels in Fig. 1(c). This requires that two-photon probability amplitude A20 must be suppressed via destructive quantum interference paths for A20 (i) |0,0Faeiφa|1,02Faeiφa|2,0, (ii) |0,0Faeiφa|1,0Fbeiφb|1,12J|2,0, (iii), |0,0Fbeiφb|0,1Faeiφa|1,12J|2,0, (iv), |0,0Fbeiφa|0,12Fbeiφb|0,22J|1,12J|2,0. Above four paths result in the destructive quantum interference to obtain the strong antibunching in the cavity mode a. So the nonlinear interaction strength U needed for destructive quantum interference becomes smaller and exactly controlled by tuning the strength ratio and relative phase between two complex driving fields.

3. Unconventional single-photon blockade

3.1. Influence of the relative phase φ on photon blockade

In Fig. 3, we show ga(2)(0) as a function of U and φ by numerically solving master Eq. (4), where Δ is given by Eq. (21), φ is plotted in units of π. In this case, the discriminant δ = m3 − 27n2 in Eq. (20) is always less than zero when phase φ varies. We set Δopt=Δopt(1) for (a), whereas Δopt=Δopt(2) for (b). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22). We notice that the unconventional single-photon blockade occurs in Figs. 3(a)–3(b) due to the quantum interference between different pathways shown in Fig. 1(c), where the Kerr nonlinear strength U is smaller than the mode broadening κ. From Fig. 3, we find the analytical results show an excellent agreement with the numerical simulation.

 

Fig. 3 Dependence of logarithmic plot of ga(2)(0) on U and φ (in units of π). The detuning Δ is given by Eq. (21). In this case, the discriminant δ = m3 − 27n2 in Eq. (20) is always less than zero when phase φ varies. We set Δopt=Δopt(1) for (a), whereas Δopt=Δopt(2) for (b). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22). Other parameters are J = 7κ, Fa = 0.3κ, r = 0.02.

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However, when the discriminant δ = m3 − 27n2 in Eq. (20) changes its sign as the strength ratio increases. How does it influence the single-photon blockade? In Figs. 4 and 5, we plot ga(2)(0) as a function of U and φ, where the detuning Δ is given by Eqs. (21), (25) and (26). Figs. 4 and 5(a)–5(d) correspond to different Δopt. The red-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). From Eqs. (21), (25) and (26), we find that there are a total of eight sets of solutions for the optimal detuning when the discriminant δ changes its sign, i.e., (Δopt(j),Δopt(k)), j = 1,2 and k = 3,4,5,6, which can be seen from Figs. 4 and 5. We notice that the conventional and unconventional single-photon blockade occurs in (a)–(d), where the Kerr nonlinear strength U is smaller and larger than the mode broadening κ, respectively.

 

Fig. 4 Dependence of logarithmic plot of ga(2)(0) on U and φ (in units of π), where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. (a) Δopt=Δopt(1) when δ < 0, whereas Δopt=Δopt(3) when δ > 0. The same for (b), (c), and (d). (b) Δopt(1) and Δopt(4), (c) Δopt(1) and Δopt(5), (d) Δopt(1) and Δopt(6). The red-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 7κ, r = 0.5, and Fa = 0.2κ.

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Fig. 5 The parameters of this figure are the same as Fig. 4, but Δopt is different. (a) Δopt(2) and Δopt(3), (b) Δopt(2) and Δopt(4), (c) Δopt(2) and Δopt(5), (d) Δopt(2) and Δopt(6). The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).

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From Fig. 4(a), we find that the optimal regimes of ga(2)(0) are not continuous, which derives from non-continuity of Δopt, i.e., it may take Δopt(1) or Δopt(3) through the change of the sign of δ. Similar observations can be found in Figs. 4 and 5. We find that Uopt is a periodic function of φ with period 2π, which can be seen from the expression of U in Eqs. (22) and (27), where U depends on φ via sin(φ), cos(φ), sin(2φ), and cos(2φ).

3.2. Comparison between exact and approximate optimal conditions

We now analyze the characteristics of photon-blockade in the coupled cavity systems by comparing the exact optimal conditions with that from the approximate one reported in [47]. The approximate optimal conditions given by [47] are obtained with these two assumptions

φ=0,
Jκ,
in which the optimal conditions are given by
ΔoptJr,
Uoptκ22Jr1r2.

Here we have set r = 1/η for comparison with [47] due to the different ratio defined in two papers.

Now we discuss the situation with the relative phase φ = 0. In Appendix B we analytically prove that three inequalities in Eqs. (23) and (24) can not hold simultaneously for all parameters. Therefore we conclude that the exact optimal detuning Δ has only two real roots, which corresponds to two optimal detuning Δopt(1),(2) in Eq. (21). Examining Eqs. (28) and (29), we can summarize the comparison of exact and approximate optimal regimes in Table 1, which shows the validity of regimes for the approximate optimal conditions. In the following, we will discuss the three regimes:

  1. When the tunnel strength J is close to κ, i.e., J ~ κ (regime I in Table 1), our exact analytical optimal conditions (21) and (22) are in good agreement with the numerical simulation, while the approximate optimal conditions (30) and (31) are not.
  2. Even if Jκ (regime II in Table 1) is satisfied, the approximate optimal conditions only give an optimal point (30) and (31), but it can not present the other optimal point, which has been given by the exact optimal values (21) and (22).
  3. When φ ≠ 0 (regime III in Table 1), the approximate optimal conditions (30) and (31) are not available in this case. In section II, we have removed the constraint of two parameters φ and J in Eqs. (28) and (29) and derived the exact optimal conditions (21)(27). Therefore there are a total of eight sets of solutions for the optimal detuning when the discriminant δ changes its sign as parameters vary, i.e., (Δopt(j),Δopt(k)), j = 1,2 and k = 3,4,5,6. When detunings Δ take its optimal values (21), (25), and (26), we can numerically find all eight optimal Kerr nonlinear strength Uopt, which can be seen from Figs. 4 and 5.

Tables Icon

Table 1. Comparison of regimes of exact and approximate optimal parameters.

Based on above key results, we concretely turn to some new aspects of our work in the following.

In order to compare with the approximate results (30) and (31), we set φ = 0. In Fig. 6 we plot ga(2)(0) as a function of U and Δ by solving numerically the master equation (4). We mark the minimum values of ga(2)(0) by r1r4 (denoted by red-stars). See Eqs. (21) and (22) [for Fig. 6(a), point r1: r1:(Uopt,Δopt(1))=(0.496κ,0.252κ); point r2:(Uopt,Δopt(2))=(0.425κ,0.1001κ)] for J = κ. When J = 20κ [Figs. 6(b) and 6(c)], point r3 takes (Uopt,Δopt(2))=(0.00861κ,0.612κ), and point r4 takes (Uopt,Δopt(1))=(0.00272κ,2.0597κ).

 

Fig. 6 Logarithmic plot for ga(2)(0) by numerically solving master Eq. (4) as a function of U and Δ at φ = 0. Parameters chosen are r = 0.1, J = κ for (a), and J = 20κ for (b) and (c). Points r1r4 (denoted by red-star) are given by Eqs. (21) and (22). while points s1 and s2 (denoted by red-circle) are given by Eqs. (30) and (31). In (d), J = κ, red-dashed line: Uopt = −0.424κ given by Δopt(2) in Eq. (22), blue-solid line: Uopt = 0.496κ given by Δopt(1) in Eq. (22); pink-dashed-dotted line: Uopt = 0.051κ given by Eq. (31).

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For the approximate optimal points s1 and s2 (denoted by red-circles) given by (30) and (31) in Fig. 6, point s1 takes (Uopt, Δopt)=(0.0505κ,0.1κ), point s2 takes (Uopt, Δopt)=(0.00253κ,2κ).

We can see from Fig. 6(a) that these two optimal values r1 and r2 obtained from (21) and (22) are consistent with the numerical simulations, but the point s1 obtained from the approximate optimal condition (30) and (31) has serious deviations with the numerical simulations. This difference comes from the assumptions Jκ failing in this case with J = κ (regime I in Table 1). Second, examining the regime of large tunnel strength Jκ, e.g., J = 20κ (regime II in Table 1), in Figs. 6(b) and 6(c) we find that the result of the approximate optimal point s2 given by Eqs. (30) and (31) is close to the exact optimal point r4. It is important to address that our exact methods give two optimal points r1 and r2 (or r3 and r4), while the approximate methods only show one point s1 (or s2). Furthermore, we plot ga(2)(0) in Fig. 6(d) (the parameter takes J = κ) and find that two exact optimal detunings Δopt(1),(2) (minimum values of blue and red-dashed lines, which correspond optimal points r1 and r2, respectively) given by (21) are consistent with numerical simulations, while the approximate solutions do not exactly predict optimal detuning (mininum value of pink dashed-dotted line, which corresponds the approximate optimal points s1).

Correspondingly, when φ ≠ 0 (regime III in Table 1), in this case there are a total of six optimal points corresponding to single-photon blockade in Fig. 2. This also is a new result of our work.

Second, we plot ga(2)(0) as a function of r and U with Δ taking its optimal values in Fig. 7, in this case J = 0.8κ (regime I in Table 1). We find that red-dashed lines given by (22) in Figs. 7(a) and 7(b) are in good agreement with those obtained by numerical simulations, but red-dashed line given by (31) in Fig. 7(c) is not consistent with those obtained by numerical simulations, which originates from the assumption of strong tunnel strength Jκ. In addition, we plot ga(2)(0) in Fig. 7(d) and find that two exact optimal detunings Δopt(1),(2) given by (21) are consistent with numerical simulations at r = 0.303, while the approximate solutions do not exactly predict optimal point at r = 0.303. To compare with φ = 0, we plot ga(2)(0) as a function of U and r in Fig. 8 by numerically solving master equation (4) when φ = 2.5rad (regime III in Table 1). We find the analytical results show an excellent agreement with the numerical simulations.

 

Fig. 7 Dependence of logarithmic plot of ga(2)(0) given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies Δopt(1) in Eq. (21) in (a), Δopt(2) in (b), and satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 0.8κ, Fa = 0.1κ. In (d), blue-solid line: the parameters take (Uopt,Δopt(1))=(2.244κ,0.291κ) given by substituting r = 0.303 into Eqs. (21) and (22); simliarly red-dashed line: (Uopt,Δopt(2))=(0.853κ,0.092κ); green-dashed-dotted line: (Uoptopt) = (0.208κ,0.242κ) given by substituting r = 0.303 into Eqs. (30) and (31).

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Fig. 8 Dependence of logarithmic plot ga(2)(0) given by numerically solving the master Eq. (4) on U and the ratio r. The detuning Δ is given by Eq. (21): (a) Δopt=Δopt(1), whereas (b) Δopt=Δopt(2) due to the discriminant δ < 0 when r varies. The red-dashed lines in (a)–(b) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 0.8κ, Fa = 0.008κ, φ = 2.5rad.

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We set same quantity J = 10κ (regime II in Table 1) in Fig. 9. Clearly, black-dashed line given by the exact optimal Kerr nonlinearity U in (22) [Figs. 9(a) and 9(b)] and approximate optimal Kerr nonlinearity in (31) [Fig. 9(c)] are in good agreement with numerical simulations. Similar to Fig. 6(c), the approximate optimal Kerr nonlinearity U decided by Eq. (31) only gives one optimal value in Fig. 9(c), but it can not predict another optimal value given by exact optimal value in Fig. 9(a). Further observation can be found in Fig. 9(d).

 

Fig. 9 Dependence of logarithmic plot of ga(2)(0) given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies Δopt(1) in Eq. (21) in (a), Δopt(2) in (b), and satisfies Eq. (30) in (c). The black-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 10κ, Fa = 0.1κ. In (d), blue-solid line: the parameters take (Uopt,Δopt(2))=(1.955κ,3.360κ) given by substituting r = 0.803 into (21) and (22); similarly green-dashed-dotted line: (Uopt,Δopt(1))=(0.115κ,8.045κ); red-dashed line: (Uoptopt) = (0.113κ,8.030κ) given by substituting r = 0.803 into Eqs. (30) and (31).

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Finally, we let J continuously varies with the strength ratio r = 0.2 in Fig. 10 (regimes I and II in Table 1). We find that good agreement between the two approaches at large tunnel strength Jκ, but in this case the result obtained by the approximate optimal conditions (30) and (31) does not work well at small J, where the red-dashed in Fig. 10(c) does not follow the exact optimal conditions. Further observation can be found in Fig. 10(d).

 

Fig. 10 Dependence of logarithmic plot of ga(2)(0) given by numerically solving the master Eq. (4) on U and J at φ = 0. The detuning Δ satisfies Δopt(1) in Eq. (21) in (a) and Δopt(2) in (b) or satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U take Eq. (31). Parameters chosen are r = 0.2, Fa = 0.1κ. In (d), blue-dashed line: the parameters take (Uopt,Δopt(1))=(0.0269κ,1.1055κ) given by substituting J = 5κ into Eqs. (21) and (22); similarly, red-solid line: (Uopt,Δopt(2))=(0.0369κ,0.247κ); black-dashed-dotted line: (Uoptopt) = (0.0208κ,κ) given by substituting J = 5κ into Eqs. (30) and (31).

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The same quantity φ = 2.5rad (regime III in Table 1) is taken in Figs. 11 and 12 comparing with Fig. 10, in which we study the influence of the tunneling rate J on the single-photon blockade effect as a function of U and J, where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when the tunnel strength J varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. The white-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). We find the analytical results show an excellent agreement with the numerical simulations.

 

Fig. 11 Dependence of logarithmic plot ga(2)(0) given by numerically solving the master Eq. (4) on U and J. The detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. (a) Δopt=Δopt(1) when δ < 0, whereas Δopt=Δopt(3) when δ > 0. The same for (b), (c), and (d). (b) Δopt(1) and Δopt(4), (c) Δopt(1) and Δopt(5), (d) Δopt(1) and Δopt(6). The white-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are r = 0.2, φ = 2.5rad, and Fa = 0.1κ.

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Fig. 12 The parameters of this figure are the same as Fig. 11, but Δopt is different. (a) Δopt(2) and Δopt(3), (b) Δopt(2) and Δopt(4), (c) Δopt(2) and Δopt(5), (b) Δopt(2) and Δopt(6). The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).

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By the analytical expressions (21)(27) for Uopt and Δopt, we can find the roles the system parameters play. Also they are used as a bench mark to check the approximate methods and to define their range of validity. For example, applying the exact optimal analytical conditions (21)(27), we can check the validity of the approximate optimal conditions (30) and (31).

Hence the exact optimal conditions (21)(27) give a fully description of photon blockade, and holds for all range of parameters. On the contrary, if there are no exact optimal conditions (21)(27), it is very difficult to experimentally find the all eight optimal regimes of photon blockade.

4. Analytical result of the equal-time second-order correlation function

Under the weak driving condition, we have |A00| ≫ |A10|, |A01| ≫ |A20|, |A11|, |A02|. Thus we assume that the vacuum state is approximately occupied with A00 ≈ 1.

Therefore, we obtain the analytical expression of ga(2)(0) by substituting Eq. (44) in Appendix C into Eq. (16)

ga(2)(0)=q2+[4rJ(κ24ΔU4Δ2)cosφ+q3]2q11t2[t1+κ2(4J2+κ28ΔU12Δ2)2],
where q1 = 16κ2Δ2 + (4J2 + κ2 − 4Δ2)2, q2 = [4r2J2κ − 8rJκ(U + 2Δ)cosφ + κ(8ΔU + 12Δ2κ2)cos(2φ) − 4rJ(κ2 − 4ΔU − 4Δ2)sinφ + 2(κ2U + 3κ2Δ − 2J2U − 4Δ2U − 4Δ3)sin(2φ)]2, q3 = [4J2U + 8Δ2(U + Δ) − 2κ2(U + 3Δ)]cos(2φ) + 4rJ(U + 2Δ)(rJ − 2κsinφ) + κ[4Δ(2U + 3Δ) − κ2]sin(2φ), t1 = [2κ2 (U + 3Δ) − 8Δ2(U + Δ) + 4J2(U + 2Δ)]2, and t2 = [(2Δcosφ + κsinφ − 2rJ)2 + (κcosφ − 2Δsinφ)2]2. We plot ga(2)(0) as functions of the detuning Δ in Fig. 13. Noticing that the discriminant δ = m3 − 27n2 in Eq. (20) changes its sign so that the solutions of Eq. (19) vary with φ. Therefore, there are a total of six optimal points corresponding to single-photon blockade, which are in good agreement with these points obtained by numerically solving the master Eq. (4) (see the minimum values in Figs. 13(a)–13(f).

 

Fig. 13 ga(2)(0) versus Δ. The solid lines and dotted lines denote the analytical expression Eq. (32) and the numerical simulation, respectively. The parameters chosen are J = 3κ, r = 0.5, and Fa = 0.03κ, φ = 4.2rad, U = 0.409κ for (a), φ = 4.2rad, U = 1.387κ for (b), φ = 0.8rad, U = 0.221κ for (c), φ = 0.8rad, U = 14.766κ for (d), φ = 0.8rad, U = 0.408κ for (e), φ = 0.8rad, U = −8.963κ for (f). We find the minimum values of the correlation functions corresponding to completely single-photon blockade are consistent with these optimal values Eqs. (21)(27), i.e., the minimum value of (a) versus Δopt(1), (b) versus Δopt(2),(c) versus Δopt(3), (d) versus Δopt(4), (e) versus Δopt(5), (f) versus Δopt(6).

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5. Analytical expression for time-delayed second-order correlation function

In this section, we will investigate the dynamical evolution of the time-delayed second-order correlation function, which can be defined as [69, 72]

ga(2)(τ)=a^(t)a^(t1)a^(t1)a^(t)a^(t)a^(t)2,
where τ = t1t,t → ∞. We now derive the analytical expression for the time-delayed second-order correlation function
ga(2)(τ)=1c3{ς3+eτκ2(ς1sin[τ(JΔ)+β1]+ς2sin[τ(J+Δ)+β2])}2+1c3{ς4+eτκ2(ς1sin[τ(JΔ)+β3]+ς2sin[τ(J+Δ)+β4])}2,
where the relevant coefficients can be found in Appendix D. For more detail derivations of Eq. (34), also see Appendix D.

In Fig. 14, ga(2)(τ) is plotted as a function of the time-delay τ. We find ga(2)(τ)=0 at τ = 0, which corresponds to single-photon blockade because U and Δ take its optimal values. As τ increases ga(2)(τ)>0, i.e., ga(2)(τ) increase above its initial value at finite delay. We thus have ga(2)(τ)>ga(2)(0). This is a violation of the classical inequality [31, 71]. This indicates that photons emitted at different times prefer to stay together comparing with initial delayed point. We find from Eq. (34) that when |Δ − J| ≫ κ/2, a fast oscillation behavior occurs for ga(2)(τ) as the delay time increases at short times. The magnitude of these osciations decreases as τ is increased and ga(2)(τ) approaches unity as τ → ∞. This explains Figs. 14(a)–14(f).

 

Fig. 14 ga(2)(τ) versus τ. U and Δ take its optimal values, i.e., (Δ,U)=(Δopt(1),Uopt(1)) or (Δopt(4),Uopt(4)). The solid lines and dotted lines denote the analytical expression (34) and the numerical simulation respectively. The parameters chosen are J = 7κ, φ = 2rad, Δopt = −3.086κ, Uopt = 7.020κ, r = 5, and Fa = 0.084κ, for (a), J = 7κ, φ = 2rad, Δopt = −2.074κ, Uopt = 6.965κ, r = 2, and Fa = 0.21κ for (b), J = 0.4κ, φ = 2rad, Δopt = −0.26κ, Uopt = 44.83κ, r = 5, and Fa = 0.0167κ for (c), J = 0.4κ, φ = 2rad, Δopt = −0.0416κ, Uopt = −0.201κ, r = 10, and Fa = 0.0832κ for (d), J = 7κ, φ = 2.7rad, Δopt = −4.52κ, Uopt = 12.247κ, r = 5, and Fa = 0.05κ for (e), J = 7κ, φ = 3.1rad, Δopt = −11.892κ, Uopt = −1.162κ, r = 5, and Fa = 0.2κ for (c).

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Interestingly, in Figs. 15(a), 15(b), and 15(d), we observe that there exists so called “beat frequency” phenomenon for photon. Let’s demonstrate the simple case at ς1ς2 = ς. Therefore from two sine waves of amplitude in Eq. (34), we can obtain,

ςsin[τ(JΔ)+β1]+ςsin[τ(J+Δ)+β2]=2ςsin(τJ+β¯1)cos(τΔ+β¯2),
where β¯1=β1+β22, β¯2=β1β22. If one frequency is much larger than another, e.g., J ≫ Δ [see Figs. 15(a) and 15(b)], the frequency of the cosine of the expression above, that is Δ, is often too slow to be perceived as a pitch. Instead, it denotes a periodic variation of the sine, whose frequency is J. Therefore, the periodic of beat frequency for photon is
Tbeat=πΔ.
Similar, when J ≪ Δ [see Fig. 15(d)], beat frequency phenomenon also can appears, whose periodic can be written as
Tbeat=πJ.
However, if the two frequencies are quite close, J ~ Δ, We can not observe beat frequency phenomenon [see line L1 in Fig. 15(c)]. It is very important to point out that the suppression of the beat can also be realized even for weak Kerr nonlinearity Uκ [see lines L2 and L3 in Fig. 15(c)]. Therefore, this may provide us a way to observe the phenomenon of photon blockade through time-delayed second-order correlation function measurements [13, 73]. In experiments we may clearly can measure time-delay second-order correlation function by measuring the beat frequency. Moreover, using this relation between beat frequency and correlation function, a bunch of other applications, such as observation [74] and control [75] of quantum beating, resonant light scattering [76], and so on, have been realized in recent years. Therefore, this scheme we obtained here is not only feasible but also extendable.

 

Fig. 15 ga(2)(τ) versus τ. The parameters chosen are φ = 3rad, r = 2, U = 2.667κ, Fa = 0.386κ, J = 10κ, Δ = 0.2κ for (a), J = 10κ, Fa = 0.278κ, Δ = 0.5κ for (b) J = 0.53κ, Fa = 0.98κ, Δ = 10κ for (d). In (c), J = 10κ, Fa = 0.0015κ, Δ = 10κ, and U = 2.667κ for L1, U = 0.1κ for L2, U = 0.02κ for L3. Interestingly, we find beats frequency for photon are observed, whose period is equal to πΔ or πJ given by Eqs. (36) and (37).

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6. Conclusions

In summary, we have studied unconventional photon-blockade effects in two linear coupled nonlinear cavities filled with nonlinear Kerr mediums with cavity modes being driven and dissipated. By analytical calculations, we derive a condition for optimized photon antibunching. We find the optimal detuning for strong photon antibunching depends on the strength ratio and the relative phase between two complex driving fields. There are a total of eight sets of solutions for the optimal detuning since the discriminant changes its sign as parameters vary. This provides us with a way to control exactly the single-photon blockade and alter it from conventional to unconventional regimes. We find that the analytical expressions of the equal-time and time-delayed correlation function are consistent with numerical simulations. Under the optimal parameters, the system can be used to generate single-photon sources.

This work shows the usefulness of the nonlinear coupled cavities system as single-photon sources on demand, with potential applications in the coherent manipulation of photon blockade with quantum photonic circuits [77]. We thus believe it is worth exactly controlling the single-photon blockade as a promising alternative to ongoing efforts in developing integrated single-photon sources by making use of quantum emitters, such as single molecules, quantum dots, defects in solids and so on.

Appendix A The optimal conditions for the situation of Δa ≠ Δb and κaκb

In this section, we cancel the constraints of the parameters Δa = Δb and κa = κb, and analytically give the optimal conditions for the situation of Δa ≠ Δb and κaκb. Therefore the condition for A02, A11, and A10 in Eq. (17) to have nontrivial solutions is that the determinant of the coefficient matrix of Eq. (17) is equal to zero, then Eq. (18) becomes

0=4eiφiJr[2i(U+Δb)+κb][2i(Δa+Δb)+κa+κb]+4r2J2[2i(U+Δa+Δb)+κa+κb]+e2iφ{8iJ2U(2iΔb+κb)[2i(U+Δb)+κb][2i(Δa+Δb)+κa+κb]}.

First, we select Δa and U as optimal parameters. After a simple calculation to Eq. (38), we obtain a general quadratic equation for the optimal detuning Δa,

η1Δa2+η2Δa+η3=0,
where the coefficients take η1=4[16J3r3sinφ+16Jrcos(φ)Δb(Jrsinφκb)4J2r2(4+cos2φ)κb8Jrsin(φ)κb2+κb3], η2=32Jrcos(φ)[J2(1+r2)+4Δb2]κb+64Jrsin(φ)Δb[J2(1+r2)+κb2]8Δbκb[4J2(1+4r2)+4Δb2+κb2]8J2r2sin(2φ)[4(J2+Δb2)+κb2], η3=64Jrcos(φ)Δb3κb16Δb4κb4Δb2{4J2[4Jrsin(φ)+κa+r2cos(2φ)κa]+[4J2(3+4r2+2r2cos2φ)+κa2]κb+2(4Jrsinφ+κa)κb2+2κb3}(κa+κb){16J3r2[J(r2+cos2φ)rsin(φ)κa]4J2r[4J(1+2r2)sinφ+rka(4+cos2φ)]κb4J(J4Jr2+2Jr2cos2φ+2rkasinφ)κb2+(8Jrsinφ+κa)κb3+κb3}+8JrΔb{Jrsin(2φ)[4J2+(κa+2κb)2]+2cos(φ)[4J2(1+r2)κb+κa2κb+κb3+2κa(J2(1+r2)+κb2)]}. The solution of Eq. (39) can be given by
Δa,opt(1),(2)=η2±η224η1η32η1.

Based on this optimal detuning Δa,opt, then the optimal Kerr nonlinearity can be expressed as

Ua,opt={8J2r2(Δa+Δb)+2cos(2φ)[4Δb2(Δa+Δb)+2Δbκaκb+(Δa+3Δb)κb2]4Jrcos(φ)[4Δb(Δa+Δb)+κb(κa+κb)]+8Jrsin(φ)[Δaκb+Δb(κa+2κb)]+sin(2φ)[8ΔaΔbκb+κb2(κa+κb)4Δb2(κa+3κb)]}/{8J2r216Jrcos(φ)(Δa+Δb)8Jrsin(φ)(κa+κb)+cos(2φ)[8J2+8Δb(Δa+Δb)2κb(κa+κb)]4sin(2φ)[Δaκb+Δb(κa+2κb)]},
where ΔaΔa,opt(1).(2) in Eq. (40).

Second, we select Δb and U as optimal parameters. Similarly, Eq. (38) becomes to,

Δb4+b1Δb3+c1Δb2+d1Δb+e1=0,
where the coefficients b1=12(Δa2rJcosφ), c1=124κb{4J2[4Jrsin(φ)+κa+r2cos(2φ)κa]+16Jrcos(φ)Δa(Jrsinφ2κb)+4Δa2κb+[4J2(3+4r2+2r2cos2φ)+κa2]κb+2(4Jrsinφ+κa)κb2+2κb3}, d1=18κb{8Jrsin(φ)Δa[J2(1+r2)+κb2]+Δaκb[4J2(1+4r2)+κb2]+J2r2sin(2φ)[4J2+4Δa2+(κa+2κb)2]2+Jrcos(φ)[4(J2+J2r2Δa2)κbκa2κbκb32κa(J2+J2r2+κb2)]}, e1=116κb{4Δa2[16J3r3sinφ4J2r2(4+cos2φ)κb8Jrsin(φ)κb2+κb3]+(κa+κb)[16J3r2[J(r2+cos2φ)rsin(φ)κa]4J2r[4J(1+2r2)sinφ+r(4+cos2φ)κa]κb4J[J4Jr2+2Jr2cos2φ+2rsin(φ)κa]κb2+(8Jrsinφ+κa)κb3+κb4]+16J2rcos(φ)Δa[2J(1+r2)κb+rsinφ(4J2+κb2)]}.

The solutions of for Δb in Eq. (42) have been given by Eqs. (21), (25), and (26) with the replacement bb1, cc1, dd1, and ee1. With the optimal detuning Δb,opt, the optimal Kerr nonlinearity Ub,opt can be given by Eq. (41), where Δb = Δb,opt. Therefore, for realistic experiment with the system parameters Δa ≠ Δb and κa ≠ κb, we can exactly control the single-photon blockade by using the optimal parameters from Eq. (39)(42).

Appendix B The proof of Eq. (19) only having two real roots when φ = 0

When the relative phase equals zero, the coefficients δ, h and η in Eqs. (23) and (24) can be reduced to

δ=J4(r21)2[N2J8+N1+16J2κ6(1+3r2)]4096,h=112[3(r21)J2κ2],η=J216(r21)[9(r21)J28κ2],
where the parameters take N1 = −162J6k2(r2−1)2 (1+3r2) − 207J4k4(r2−1)2 + 64κ8, N2 = −216r2(r2−1)2 (1+r2). When the discrimination δ > 0 in Eq. (23), if and only if two inequalities (24) are simultaneously satisfied, Eq. (19) has four real roots, otherwise it has four conjugate complex roots. We now solve the three inequalities in Eqs. (23) and (24), we find three inequalities can not hold simultaneously for all parameters. Therefore, there are nonphysical optimal values when discriminant δ > 0 at φ = 0. We conclude that the exact optimal detuning Δ has only real roots Δopt(1),(2) in Eq. (21) at φ = 0.

Appendix C Analytical expression of the probability amplitude

We obtain the analytical expression for the equal-time second-order correlation function by solving the steady state in Eqs. (7)(12) in weak driving limit

A¯10=2eiφbFa[2rJ+ieiφ(κ+2iΔ)]4J2+(κ+2iΔ)2,A¯20=22e2iφbFa2(4irJw1eiφ+w2+e2iφw3)w4[(κ+2iΔ)2w5+4J2κ+4J2(U+2Δ)i],A¯11=4e2iφbFa2(2Jeiφ+irκ2rΔ)(iw1eiφ+w22rJ)w4[(κ+2iΔ)2w5+4J2κ+4J2(U+2Δ)i],
where w1 = (κ + 2iΔ)[κ + 2i(U + Δ)], w2 = 4r2J2[κ + i(U + 2Δ)], w3 = 4iJ2U−(κ + 2iΔ)2[κ + 2i(U + Δ)], w4 = 4J2 + (κ + 2iΔ)2, w5 = κ + 2i(U + Δ).

Appendix D Analytical derivation of the time-delayed second-order correlation function

Applying the Born approximation and assuming the environment being in zero temperature to Eq. (33), we can obtain the analytical expression for time-delayed two-order correlation function [38],

ga(2)(τ)=TrS[A^A^ρ˜(τ)],
with
ρ˜(τ)=TrE[UT(τ)ρ˜(0)ρEUT(τ)],
where the new initial state is defined by ρ˜(0)=a^ρ¯a^ and A^=a^/n¯a, ρ¯ is the steady state for Eq. (4). The unitary evolution operator is given by UT(t)=exp(iH^Tt), H^T=H^+H^E+H^I, and H^ given by Eq. (3), H^E and H^I the Hamiltonians of the environment and the system-environment interaction, respectively. ρE is the vacuum state of the environment. The dynamical evolution for Eqs. (46) and (4) are equal, i.e., ρ˜˙(τ)ρ˙(τ) but they have different initial state. Therefore under effective Hamiltonian approximation, ρ˜(τ)=|Ψ˜(τ)Ψ˜(τ)| and ρ(τ) = |Ψ(τ)〉〈Ψ(τ)| are equivalent to |Ψ˜˙(τ)=|Ψ˙(τ). Therefore the time-delayed correlation function can be calculated by introducing a initial state
|Ψ˜(0)=a^|Ψ¯,
with |Ψ¯ given by the steady state solutions of Eqs. (7)(12). Under this initial state, we obtain the time-delayed second-order correlation function,
ga(2)(τ)=|A10(τ)|2|A¯10|4.

To be specific, the initial state after the annihilation of a photon in the left cavity a is

|Ψ˜(0)=A¯10|00+2A¯20|10+A¯11|01.

Equation (48) can be obtained by solving Eqs. (7)(9) for these amplitudes with the initial state (49). Under the weak driving condition, we have A00=A¯10. Therefore, Eq. (7) decouples with Eqs. (8) and (9). Defining two quantities

A10(τ)+A01(τ)=x(τ),A10(τ)A01(τ)=y(τ),
and substituting Eq. (50) into Eqs. (8) and (9), we obtain
xτ+u1x(τ)=λ1,yτ+u2y(τ)=λ2,
where we ignore the high-order terms A11, A20 and A20, and the initial values x(0)=2A¯20+A¯11, y(0)=2A¯20A¯11, decided by Eq. (44). The other parameters take u1 = i(δa + J), u2 = i(δaJ), λ1=iFaeiφb(eiφ+r)A00, λ2=iFaeiφb(eiφr)A00 with A00Ā10. The solutions of the first order ordinary differential equation (51) are
x(τ)=x(0)eu1τ+λ1u1(1eu1τ),y(τ)=y(0)eu2τ+λ2u2(1eu2τ).

Noticing Eq. (44), Eq. (34) can be obtained by substituting Eqs. (50) and (52) into Eq. (48), where these time-independent coefficients take c3 = 64[(−2aJ + 2Δcosφ + κsinφ)2 + (κcosφ − 2Δsinφ)2]2/[16κ2Δ2 + (4J2 + κ2 − 4Δ2)2]2, r1 = 2[−(2A001(J + Δ) − A002κ)(r + cosφ) + (2A002(J + Δ) + A001κ)sinφ]/[4(J + Δ)2 + κ2], r2 = −2[(2A002(J + Δ) + A001κ)(r + cosφ) + (2A001(J + Δ) − A002κ)sinφ]/[4(J + Δ)2 + κ2], r3 = 2[(2A001(J − Δ) + A002κ)(cosφr) + (2A002(Δ − J) + A001κ)sinφ]/[4(J − Δ)2 + κ2], r4 = 2[(2A002(J − Δ) − A001κ)(cosφr) + (2A001(J − Δ) + A002κ)sinφ]/[4(J −Δ)2 + κ2]. x01=Re{x(0)/[Fa2e2iφb]}, y01=Im{x(0)/[Fa2e2iφb]}, x02=Re{y(0)/[Fa2e2iφb]}, y02=Im{y(0)/[Fa2e2iφb]}, A001=Re{A¯10/[Faeiφb]}, A002=Im{A¯10/[Faeiφb]}, z1 = y02r4, z2 = y01r2, z3 = x02r3, z4 = r1x01, ς1=z12+z32, ς2=z22+z42, ς3 = r2 + r4, ς4 = r1 + r3, β1=arctan(z1z3), β2=arctan(z2z4), β3=arctan(z3z1), β4=arctan(z4z2).

Acknowledgments

We would like to thank Prof. Yong Li and Prof. Xiao-Qiang Shao for valuable discussions. This work is supported by National Natural Science Foundation of China (NSFC) under grant Nos 11175032, 61475033, 11405008, 11405026 and the Plan for Scientific and Technological Development of Jilin Province (No. 20150520083JH).

References and links

1. L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996). [CrossRef]  

2. A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1, 215–223 (2007). [CrossRef]  

3. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009). [CrossRef]  

4. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011). [CrossRef]  

5. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef]   [PubMed]  

6. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007). [CrossRef]  

7. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005). [CrossRef]   [PubMed]  

8. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008). [CrossRef]   [PubMed]  

9. F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010). [CrossRef]  

10. M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013). [CrossRef]  

11. J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015). [CrossRef]   [PubMed]  

12. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]  

13. C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov Jr., M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011). [CrossRef]   [PubMed]  

14. A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011). [CrossRef]   [PubMed]  

15. A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997). [CrossRef]  

16. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011). [CrossRef]   [PubMed]  

17. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). [CrossRef]   [PubMed]  

18. J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010). [CrossRef]  

19. J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013). [CrossRef]  

20. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013). [CrossRef]  

21. X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015). [CrossRef]  

22. W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015). [CrossRef]  

23. X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013). [CrossRef]   [PubMed]  

24. L. Qiu, L. Gan, W. Ding, and Z. Y. Li, “Single-photon generation by pulsed laser in optomechanical system via photon blockade effect,” J. Opt. Soc. Am. B 30, 1683–1687 (2013). [CrossRef]  

25. X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015). [CrossRef]   [PubMed]  

26. H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015). [CrossRef]  

27. A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013). [CrossRef]  

28. M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010). [CrossRef]  

29. M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999). [CrossRef]  

30. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992). [CrossRef]   [PubMed]  

31. R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999). [CrossRef]  

32. J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999). [CrossRef]  

33. I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002). [CrossRef]   [PubMed]  

34. S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002). [CrossRef]  

35. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007). [CrossRef]  

36. Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014). [CrossRef]  

37. A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013). [CrossRef]  

38. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014). [CrossRef]  

39. D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007). [CrossRef]  

40. D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009). [CrossRef]  

41. F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014). [CrossRef]   [PubMed]  

42. E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014). [CrossRef]  

43. T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010). [CrossRef]   [PubMed]  

44. M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998). [CrossRef]  

45. Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010). [CrossRef]  

46. M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011). [CrossRef]  

47. X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014). [CrossRef]  

48. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013). [CrossRef]  

49. X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014). [CrossRef]  

50. J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007). [CrossRef]   [PubMed]  

51. X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014). [CrossRef]  

52. T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013). [CrossRef]  

53. H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013). [CrossRef]  

54. D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014). [CrossRef]  

55. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015). [CrossRef]  

56. Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015). [CrossRef]  

57. S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013). [CrossRef]  

58. O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014). [CrossRef]  

59. M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011). [CrossRef]  

60. M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014). [CrossRef]  

61. A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012). [CrossRef]   [PubMed]  

62. W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014). [CrossRef]  

63. X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

64. V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

65. O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014). [CrossRef]  

66. T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012). [CrossRef]  

67. H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

68. S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012). [CrossRef]  

69. A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006). [CrossRef]  

70. R. Loudon, The Quantum Theory of Light (Oxford University, 2003).

71. H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991). [CrossRef]  

72. S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010). [CrossRef]  

73. J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009). [CrossRef]   [PubMed]  

74. Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988). [CrossRef]   [PubMed]  

75. A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013). [CrossRef]  

76. M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014). [CrossRef]  

77. J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]  

References

  • View by:
  • |
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  • |

  1. L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
    [Crossref]
  2. A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1, 215–223 (2007).
    [Crossref]
  3. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
    [Crossref]
  4. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
    [Crossref]
  5. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
    [Crossref] [PubMed]
  6. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [Crossref]
  7. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
    [Crossref] [PubMed]
  8. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
    [Crossref] [PubMed]
  9. F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
    [Crossref]
  10. M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
    [Crossref]
  11. J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
    [Crossref] [PubMed]
  12. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
    [Crossref]
  13. C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
    [Crossref] [PubMed]
  14. A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
    [Crossref] [PubMed]
  15. A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
    [Crossref]
  16. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
    [Crossref] [PubMed]
  17. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
    [Crossref] [PubMed]
  18. J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
    [Crossref]
  19. J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
    [Crossref]
  20. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
    [Crossref]
  21. X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
    [Crossref]
  22. W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
    [Crossref]
  23. X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
    [Crossref] [PubMed]
  24. L. Qiu, L. Gan, W. Ding, and Z. Y. Li, “Single-photon generation by pulsed laser in optomechanical system via photon blockade effect,” J. Opt. Soc. Am. B 30, 1683–1687 (2013).
    [Crossref]
  25. X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
    [Crossref] [PubMed]
  26. H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
    [Crossref]
  27. A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
    [Crossref]
  28. M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010).
    [Crossref]
  29. M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999).
    [Crossref]
  30. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
    [Crossref] [PubMed]
  31. R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
    [Crossref]
  32. J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
    [Crossref]
  33. I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
    [Crossref] [PubMed]
  34. S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
    [Crossref]
  35. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
    [Crossref]
  36. Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
    [Crossref]
  37. A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013).
    [Crossref]
  38. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
    [Crossref]
  39. D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
    [Crossref]
  40. D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
    [Crossref]
  41. F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
    [Crossref] [PubMed]
  42. E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
    [Crossref]
  43. T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
    [Crossref] [PubMed]
  44. M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
    [Crossref]
  45. Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
    [Crossref]
  46. M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
    [Crossref]
  47. X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
    [Crossref]
  48. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
    [Crossref]
  49. X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014).
    [Crossref]
  50. J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
    [Crossref] [PubMed]
  51. X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
    [Crossref]
  52. T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
    [Crossref]
  53. H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
    [Crossref]
  54. D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
    [Crossref]
  55. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
    [Crossref]
  56. Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
    [Crossref]
  57. S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
    [Crossref]
  58. O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
    [Crossref]
  59. M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
    [Crossref]
  60. M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
    [Crossref]
  61. A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
    [Crossref] [PubMed]
  62. W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
    [Crossref]
  63. X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).
  64. V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).
  65. O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
    [Crossref]
  66. T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012).
    [Crossref]
  67. H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).
  68. S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
    [Crossref]
  69. A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
    [Crossref]
  70. R. Loudon, The Quantum Theory of Light (Oxford University, 2003).
  71. H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
    [Crossref]
  72. S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
    [Crossref]
  73. J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
    [Crossref] [PubMed]
  74. Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988).
    [Crossref] [PubMed]
  75. A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
    [Crossref]
  76. M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
    [Crossref]
  77. J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
    [Crossref]

2015 (7)

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
[Crossref] [PubMed]

X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
[Crossref]

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

2014 (13)

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

2013 (13)

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

L. Qiu, L. Gan, W. Ding, and Z. Y. Li, “Single-photon generation by pulsed laser in optomechanical system via photon blockade effect,” J. Opt. Soc. Am. B 30, 1683–1687 (2013).
[Crossref]

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
[Crossref]

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

2012 (3)

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012).
[Crossref]

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
[Crossref]

2011 (7)

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

2010 (6)

J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010).
[Crossref]

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

2009 (4)

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

2008 (2)

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

2007 (5)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1, 215–223 (2007).
[Crossref]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

2006 (1)

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

2005 (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

2002 (2)

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
[Crossref]

2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

1999 (3)

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999).
[Crossref]

1998 (1)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

1997 (1)

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

1996 (1)

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

1992 (1)

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
[Crossref] [PubMed]

1991 (1)

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

1988 (1)

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988).
[Crossref] [PubMed]

Abdumalikov, A. A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Andreani, L. C.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Angelakis, D. G.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

Aoki, T.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

Ashhab, S.

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Asmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Auffèves, A.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Auffovès, A.

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

Aumentado, J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Bachor, H. A.

H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

Bajcsy, M.

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

Bajer, J.

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Bamba, M.

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

Barberis-Blostein, P.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

Barros, H. G.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Baur, M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Bayer, M.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Becher, C.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Bennett, S. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Bensen, O.

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

Berstermann, T.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Blais, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Blatt, R.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Boca, A.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Boozer, A. D.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Børkje, K.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

Bose, S.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

Bozyigit, D.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Brecha, R. J.

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

Carmichael, H. J.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
[Crossref] [PubMed]

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Chang, D. E.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Cheng, J.

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

Cimmarusti, A. D.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
[Crossref]

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

Clerk, A. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

da Silva, M. P.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Davidovich, L.

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

Davis, C. C.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

Dayan, B.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

Demler, E. A.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Deutsch, M.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Didier, N.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

Ding, W.

Donegan, J. F.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[Crossref]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Dremin, A. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Dubin, F.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Eichler, C.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Englund, D.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Fan, S. H.

J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

Faraon, A.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Fazio, R.

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Ferretti, S.

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Filipp, S.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Fink, J. M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Flayac, H.

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

Forchel, A.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Fratini, F.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Furusawa, A.

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Fushman, I.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Gan, L.

Geng, W. D.

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
[Crossref] [PubMed]

Gerace, D.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Gies, C.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Girvin, S. M.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

Gu, X.

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

Gullans, M.

X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
[Crossref]

Gungor, A.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

Gutbrod, T.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Habraken, S. J. M.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Hartmann, M. J.

M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010).
[Crossref]

Hoffman, A. J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Höfling, S.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Hommel, D.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Houck, A. A.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Imamo?lu, A.

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999).
[Crossref]

Imamoglu, A.

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Jahnke, F.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Jing, H.

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

Johansson, J. R.

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

Kalden, J.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Kan, H.

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

Kim, J.

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

Kimble, H. J.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Kistner, C.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

Knipp, P. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Kómár, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Konthasinghe, K.

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

Kruse, C.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Kulakovskii, V. D.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Kyriienko, O.

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

Lang, C.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Law, C. K.

J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

Leib, M.

M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010).
[Crossref]

Lemonde, M. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

Li, Y.

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014).
[Crossref]

Li, Y. J.

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

Li, Z. Y.

Liao, J. Q.

J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

Liew, T. C. H.

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
[Crossref]

T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

Liu, J. Y.

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

Liu, Y. M.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Liu, Y. X.

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 2003).

Lü, X. Y

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Lü, X. Y.

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

Lukin, M. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Lütkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

Majumdar, A.

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

Mandel, L.

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988).
[Crossref] [PubMed]

Mascarenhas, E.

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

Miller, R.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Miranowicz, A.

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Montangero, S.

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

Muller, A.

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Niu, Z. C.

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

Nori, F.

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Northup, T. E.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Nunnenkamp, A.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

ÓBrien, J. L.

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Orozco, L. A.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

Ostby, E. P.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988).
[Crossref] [PubMed]

Paprzycka, M.

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

Parkins, A. S.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
[Crossref]

Patterson, B. D.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Peiris, M.

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

Peng, Y. W.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Petroff, P.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Poizat, J-Ph.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Qiu, L.

Rabl, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

Rakovich, Y. P.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[Crossref]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

Rebic, S.

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
[Crossref]

Reinecke, T. L.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Reithmaier, J. P.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Reitzenstein, S.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Rice, P. R.

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

Rundquist, A.

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

Russo, C.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Safari, L.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Santos, M. F.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

Savona, V.

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
[Crossref]

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Schmidt, H.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Schmidt, P. O.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Schmidt, S.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Schneider, C.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Schroeder, C. A.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

Shelykh, I. A.

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

Shen, H. Z.

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

Shen, J. T.

J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

Shields, A. J.

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1, 215–223 (2007).
[Crossref]

Smolyaninov, I. I.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

Sorensen, A. S.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Spietz, L.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Srinivasan, S. J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

Stannigel, K.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Steffen, L.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Stoltz, N.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Stute, A.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Tan, S. M.

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
[Crossref]

Tang, J.

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
[Crossref] [PubMed]

Taylor, J. M.

X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
[Crossref]

Tian, L.

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
[Crossref] [PubMed]

Tureci, H. E.

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Türeci, H. E.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Vahala, K. J.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

Valente, D.

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Verger, A.

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

Vuckovic, J.

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Wallraff, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Wang, H.

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

Werner, M. J.

M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999).
[Crossref]

Wiersig, J.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Woods, G.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Wu, Y.

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Xiao, M.

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

Xu, X. L.

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
[Crossref] [PubMed]

Xu, X. N.

X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
[Crossref]

Xu, X. W.

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

Yamamoto, Y.

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

Yi, X. X.

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

Yu, Y.

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

Yu, Z. Y.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Zayats, A. V.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

Zhang, J.

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

Zhang, W.

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

Zhang, W. M.

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Zhang, W. Z.

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

Zhou, L.

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

Zhou, Y. H.

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

Zoller, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Appl. Phys. Lett. (1)

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
[Crossref]

Europhys. Lett. (1)

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

J. Opt. B: At. Mol. Opt. Phys. (1)

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[Crossref]

Nat. Photonics (3)

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1, 215–223 (2007).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Nat. Phys. (4)

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Nature (4)

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

New J. Phys. (5)

M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010).
[Crossref]

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
[Crossref]

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

Opt. Commun. (1)

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

Phys. Rev. A (28)

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014).
[Crossref]

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
[Crossref]

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999).
[Crossref]

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
[Crossref] [PubMed]

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
[Crossref]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

Phys. Rev. B (4)

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
[Crossref]

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

Phys. Rev. Lett. (13)

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988).
[Crossref] [PubMed]

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref] [PubMed]

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

Rev. Mod. Phys. (4)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

Sci. Rep. (2)

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
[Crossref] [PubMed]

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Science (1)

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

Other (3)

H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

V. Savona, “Unconventional photon blockade in coupled optomechanical systems,” arXiv:1302.5937 (2013).

R. Loudon, The Quantum Theory of Light (Oxford University, 2003).

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Figures (15)

Fig. 1
Fig. 1 (a) Scheme diagram of the setup. Two cavities are driven by external fields with the complex Rabi frequency F a e i φ a and F b e i φ b, respectively. (b) The equal-time second-order correlation function g a ( 2 ) ( 0 ) is plotted by numerically solving the master Eq. (4). The nearly perfect antibunching can be obtained at U = 1.593 × 10−3κ with φ = 1.6 rad. The other parameters are chosen as Δa = Δb = 0.1539κ, J = 7κ, Ua = 0, and Fa = 0.7κ, Fb = 0.028κ, φ = 1.6rad for blue-bold line, φ = 1.8rad for dashed-green line, φ = 1.4rad for red-dashed-dotted line. (c) The first (second) number in |mn〉 denotes the photon number in cavity a (cavity b). The transition paths (two dashed lines with arrows) lead to the quantum interference (indicated by the ellipse) for the strong photon antibunching.
Fig. 2
Fig. 2 Dependence of logarithmic plot for g a ( 2 ) ( 0 ) on U (UUb) and Δ by numerically solving master Eq. (4). Defining FbrFa. Parameters chosen are J = 7κ, Fa = 0.7κ, Ua = 0, and r = 0.02, φ = 2rad for (a), and r = 0.2, φ = 1rad for(b)–(d). We have an interesting observation that there are two and four minimum values for two-order correlation function for (a) [see points P1 and P2 denoted by blue-star] and (b)–(d) [see points P3-P6 denoted by blue-star], respectively. Therefore points P1-P4 correspond to unconventional single-photon blockade, which is smaller than the mode broadening κ, while points P5 and P6 indicate conventional single-photon blockade, which is larger than the mode broadening κ.
Fig. 3
Fig. 3 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) on U and φ (in units of π). The detuning Δ is given by Eq. (21). In this case, the discriminant δ = m3 − 27n2 in Eq. (20) is always less than zero when phase φ varies. We set Δ o p t = Δ o p t ( 1 ) for (a), whereas Δ o p t = Δ o p t ( 2 ) for (b). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22). Other parameters are J = 7κ, Fa = 0.3κ, r = 0.02.
Fig. 4
Fig. 4 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) on U and φ (in units of π), where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. (a) Δ o p t = Δ o p t ( 1 ) when δ < 0, whereas Δ o p t = Δ o p t ( 3 ) when δ > 0. The same for (b), (c), and (d). (b) Δ o p t ( 1 ) and Δ o p t ( 4 ), (c) Δ o p t ( 1 ) and Δ o p t ( 5 ), (d) Δ o p t ( 1 ) and Δ o p t ( 6 ). The red-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 7κ, r = 0.5, and Fa = 0.2κ.
Fig. 5
Fig. 5 The parameters of this figure are the same as Fig. 4, but Δopt is different. (a) Δ o p t ( 2 ) and Δ o p t ( 3 ), (b) Δ o p t ( 2 ) and Δ o p t ( 4 ), (c) Δ o p t ( 2 ) and Δ o p t ( 5 ), (d) Δ o p t ( 2 ) and Δ o p t ( 6 ). The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).
Fig. 6
Fig. 6 Logarithmic plot for g a ( 2 ) ( 0 ) by numerically solving master Eq. (4) as a function of U and Δ at φ = 0. Parameters chosen are r = 0.1, J = κ for (a), and J = 20κ for (b) and (c). Points r1r4 (denoted by red-star) are given by Eqs. (21) and (22). while points s1 and s2 (denoted by red-circle) are given by Eqs. (30) and (31). In (d), J = κ, red-dashed line: Uopt = −0.424κ given by Δ o p t ( 2 ) in Eq. (22), blue-solid line: Uopt = 0.496κ given by Δ o p t ( 1 ) in Eq. (22); pink-dashed-dotted line: Uopt = 0.051κ given by Eq. (31).
Fig. 7
Fig. 7 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies Δ o p t ( 1 ) in Eq. (21) in (a), Δ o p t ( 2 ) in (b), and satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 0.8κ, Fa = 0.1κ. In (d), blue-solid line: the parameters take ( U o p t , Δ o p t ( 1 ) ) = ( 2.244 κ , 0.291 κ ) given by substituting r = 0.303 into Eqs. (21) and (22); simliarly red-dashed line: ( U o p t , Δ o p t ( 2 ) ) = ( 0.853 κ , 0.092 κ ); green-dashed-dotted line: (Uoptopt) = (0.208κ,0.242κ) given by substituting r = 0.303 into Eqs. (30) and (31).
Fig. 8
Fig. 8 Dependence of logarithmic plot g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and the ratio r. The detuning Δ is given by Eq. (21): (a) Δ o p t = Δ o p t ( 1 ), whereas (b) Δ o p t = Δ o p t ( 2 ) due to the discriminant δ < 0 when r varies. The red-dashed lines in (a)–(b) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 0.8κ, Fa = 0.008κ, φ = 2.5rad.
Fig. 9
Fig. 9 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies Δ o p t ( 1 ) in Eq. (21) in (a), Δ o p t ( 2 ) in (b), and satisfies Eq. (30) in (c). The black-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 10κ, Fa = 0.1κ. In (d), blue-solid line: the parameters take ( U o p t , Δ o p t ( 2 ) ) = ( 1.955 κ , 3.360 κ ) given by substituting r = 0.803 into (21) and (22); similarly green-dashed-dotted line: ( U o p t , Δ o p t ( 1 ) ) = ( 0.115 κ , 8.045 κ ); red-dashed line: (Uoptopt) = (0.113κ,8.030κ) given by substituting r = 0.803 into Eqs. (30) and (31).
Fig. 10
Fig. 10 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and J at φ = 0. The detuning Δ satisfies Δ o p t ( 1 ) in Eq. (21) in (a) and Δ o p t ( 2 ) in (b) or satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U take Eq. (31). Parameters chosen are r = 0.2, Fa = 0.1κ. In (d), blue-dashed line: the parameters take ( U o p t , Δ o p t ( 1 ) ) = ( 0.0269 κ , 1.1055 κ ) given by substituting J = 5κ into Eqs. (21) and (22); similarly, red-solid line: ( U o p t , Δ o p t ( 2 ) ) = ( 0.0369 κ , 0.247 κ ); black-dashed-dotted line: (Uoptopt) = (0.0208κ,κ) given by substituting J = 5κ into Eqs. (30) and (31).
Fig. 11
Fig. 11 Dependence of logarithmic plot g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and J. The detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. (a) Δ o p t = Δ o p t ( 1 ) when δ < 0, whereas Δ o p t = Δ o p t ( 3 ) when δ > 0. The same for (b), (c), and (d). (b) Δ o p t ( 1 ) and Δ o p t ( 4 ), (c) Δ o p t ( 1 ) and Δ o p t ( 5 ), (d) Δ o p t ( 1 ) and Δ o p t ( 6 ). The white-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are r = 0.2, φ = 2.5rad, and Fa = 0.1κ.
Fig. 12
Fig. 12 The parameters of this figure are the same as Fig. 11, but Δopt is different. (a) Δ o p t ( 2 ) and Δ o p t ( 3 ), (b) Δ o p t ( 2 ) and Δ o p t ( 4 ), (c) Δ o p t ( 2 ) and Δ o p t ( 5 ), (b) Δ o p t ( 2 ) and Δ o p t ( 6 ). The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).
Fig. 13
Fig. 13 g a ( 2 ) ( 0 ) versus Δ. The solid lines and dotted lines denote the analytical expression Eq. (32) and the numerical simulation, respectively. The parameters chosen are J = 3κ, r = 0.5, and Fa = 0.03κ, φ = 4.2rad, U = 0.409κ for (a), φ = 4.2rad, U = 1.387κ for (b), φ = 0.8rad, U = 0.221κ for (c), φ = 0.8rad, U = 14.766κ for (d), φ = 0.8rad, U = 0.408κ for (e), φ = 0.8rad, U = −8.963κ for (f). We find the minimum values of the correlation functions corresponding to completely single-photon blockade are consistent with these optimal values Eqs. (21)(27), i.e., the minimum value of (a) versus Δ o p t ( 1 ), (b) versus Δ o p t ( 2 ),(c) versus Δ o p t ( 3 ), (d) versus Δ o p t ( 4 ), (e) versus Δ o p t ( 5 ), (f) versus Δ o p t ( 6 ).
Fig. 14
Fig. 14 g a ( 2 ) ( τ ) versus τ. U and Δ take its optimal values, i.e., ( Δ , U ) = ( Δ o p t ( 1 ) , U o p t ( 1 ) ) or ( Δ o p t ( 4 ) , U o p t ( 4 ) ). The solid lines and dotted lines denote the analytical expression (34) and the numerical simulation respectively. The parameters chosen are J = 7κ, φ = 2rad, Δopt = −3.086κ, Uopt = 7.020κ, r = 5, and Fa = 0.084κ, for (a), J = 7κ, φ = 2rad, Δopt = −2.074κ, Uopt = 6.965κ, r = 2, and Fa = 0.21κ for (b), J = 0.4κ, φ = 2rad, Δopt = −0.26κ, Uopt = 44.83κ, r = 5, and Fa = 0.0167κ for (c), J = 0.4κ, φ = 2rad, Δopt = −0.0416κ, Uopt = −0.201κ, r = 10, and Fa = 0.0832κ for (d), J = 7κ, φ = 2.7rad, Δopt = −4.52κ, Uopt = 12.247κ, r = 5, and Fa = 0.05κ for (e), J = 7κ, φ = 3.1rad, Δopt = −11.892κ, Uopt = −1.162κ, r = 5, and Fa = 0.2κ for (c).
Fig. 15
Fig. 15 g a ( 2 ) ( τ ) versus τ. The parameters chosen are φ = 3rad, r = 2, U = 2.667κ, Fa = 0.386κ, J = 10κ, Δ = 0.2κ for (a), J = 10κ, Fa = 0.278κ, Δ = 0.5κ for (b) J = 0.53κ, Fa = 0.98κ, Δ = 10κ for (d). In (c), J = 10κ, Fa = 0.0015κ, Δ = 10κ, and U = 2.667κ for L1, U = 0.1κ for L2, U = 0.02κ for L3. Interestingly, we find beats frequency for photon are observed, whose period is equal to π Δ or π J given by Eqs. (36) and (37).

Tables (1)

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Table 1 Comparison of regimes of exact and approximate optimal parameters.

Equations (52)

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H ^ 0 = h ¯ ω a a ^ a ^ + h ¯ ω b b ^ b ^ + h ¯ J ( a ^ b ^ + b ^ a ^ ) + U a a ^ a ^ a ^ a ^ + U b b ^ b ^ b ^ b ^ ,
H ^ d r ( t ) = H ^ 0 + ( h ¯ F a a ^ e i φ a i ω L t + h ¯ F b b ^ e i φ b i ω L t + H . c . ) ,
H ^ = h ¯ Δ a a ^ a ^ + h ¯ Δ b b ^ b ^ + h ¯ J ( a ^ b ^ + a ^ b ^ ) + U a a ^ a ^ a ^ a ^ + U b b ^ b ^ b ^ b ^ + ( h ¯ F a e i φ a a ^ + h ¯ F b e i φ b b ^ + H . c . ) ,
ρ ˙ = i [ H ^ , ρ ] + κ a D ( a ^ ) ρ + κ b D ( b ^ ) ρ ,
g a ( 2 ) ( 0 ) = a ^ 2 a ^ 2 a ^ a ^ 2 .
| Ψ = A 00 | 00 + A 10 | 10 + A 01 | 01 + A 20 | 20 + A 02 | 02 + A 11 | 11 ,
i t A 00 = F a e i φ a A 10 + F b e i φ b A 01 ,
i t A 10 = δ a A 10 + J A 01 + F a e i φ a A 00 + A b e i φ b A 11 + 2 F a e i φ a A 20 ,
i t A 01 = δ b A 01 + J A 10 + F b e i φ b A 00 + F a e i φ a A 11 + 2 F b e i φ b A 02 ,
i t A 11 = ( δ a + δ b ) A 11 + 2 J ( A 20 + A 02 ) + F b e i φ b A 10 + F a e i φ a A 01 ,
i t A 20 = 2 ( δ a + U a ) A 20 + 2 J A 11 + 2 F a e i φ a A 10 ,
i t A 02 = 2 ( δ b + U b ) A 02 + 2 J A 11 + 2 F b e i φ b A 01 ,
δ a A 10 + J A 01 = F a e i φ a A 00 , J A 10 + δ b A 01 = F b e i φ b A 00 ,
0 = 2 ( δ a + U a ) A 20 + 2 J A 11 + 2 F a e i φ a A 10 , 0 = 2 ( δ b + U b ) A 02 + 2 J A 11 + 2 F b e i φ b A 01 , 0 = ( δ a + δ b ) A 11 + 2 J ( A 20 + A 02 ) + F b e i φ b A 10 + F a e i φ a A 01 .
A 01 A 10 = e i φ a F a J e i φ b F b δ a e i φ b F b J e i φ a F a δ b ζ .
g a ( 2 ) ( 0 ) 2 | A 20 | 2 | A 10 | 4 .
0 = 2 J A 11 + 2 F a e i φ a A 10 , 0 = 2 ( δ b + U b ) A 02 + 2 J A 11 + 2 F b e i φ b ζ A 10 , 0 = ( δ a + δ b ) A 11 + 2 J A 02 + F b e i φ b A 10 + F a e i φ a ζ A 10 .
0 = 4 i r 2 J 2 [ κ + i ( U + 2 Δ ) ] + e 2 i φ [ ( κ + 2 i Δ ) 2 ( 2 U + 2 Δ i κ ) 4 J 2 U ] 4 r e i φ J ( κ + 2 i Δ ) [ κ + 2 i ( U + Δ ) ] .
Δ 4 + 4 b Δ 3 + 6 c Δ 2 + 4 d Δ + e = 0 ,
δ = m 3 27 n 2 < 0
Δ o p t ( 1 ) , ( 2 ) = [ b sgn ( g ) ] λ ± | g | / λ λ + 3 h ,
U o p t = 4 r J v 2 + 2 Δ ( 3 κ 2 4 Δ 2 ) sin ( 2 φ ) + v 3 4 cos ( φ ) v 1 8 Δ [ κ cos ( 2 φ ) + 2 r J sin ( φ ) ] ,
δ = m 3 27 n 2 > 0 ,
h > 0 , η > 0 ,
Δ o p t ( 3 ) , ( 4 ) = b + s γ 1 ± γ 2 ± γ 3 ,
Δ o p t ( 5 ) , ( 6 ) = b s γ 1 ± γ 2 γ 3 .
U o p t = 4 r J v 2 + 2 Δ ( 3 κ 2 4 Δ 2 ) sin ( 2 φ ) + v 3 4 cos ( φ ) v 1 8 Δ [ κ cos ( 2 φ ) + 2 r J sin ( φ ) ] ,
φ = 0 ,
J κ ,
Δ o p t J r ,
U o p t κ 2 2 J r 1 r 2 .
g a ( 2 ) ( 0 ) = q 2 + [ 4 r J ( κ 2 4 Δ U 4 Δ 2 ) cos φ + q 3 ] 2 q 1 1 t 2 [ t 1 + κ 2 ( 4 J 2 + κ 2 8 Δ U 12 Δ 2 ) 2 ] ,
g a ( 2 ) ( τ ) = a ^ ( t ) a ^ ( t 1 ) a ^ ( t 1 ) a ^ ( t ) a ^ ( t ) a ^ ( t ) 2 ,
g a ( 2 ) ( τ ) = 1 c 3 { ς 3 + e τ κ 2 ( ς 1 sin [ τ ( J Δ ) + β 1 ] + ς 2 sin [ τ ( J + Δ ) + β 2 ] ) } 2 + 1 c 3 { ς 4 + e τ κ 2 ( ς 1 sin [ τ ( J Δ ) + β 3 ] + ς 2 sin [ τ ( J + Δ ) + β 4 ] ) } 2 ,
ς sin [ τ ( J Δ ) + β 1 ] + ς sin [ τ ( J + Δ ) + β 2 ] = 2 ς sin ( τ J + β ¯ 1 ) cos ( τ Δ + β ¯ 2 ) ,
T b e a t = π Δ .
T b e a t = π J .
0 = 4 e i φ i J r [ 2 i ( U + Δ b ) + κ b ] [ 2 i ( Δ a + Δ b ) + κ a + κ b ] + 4 r 2 J 2 [ 2 i ( U + Δ a + Δ b ) + κ a + κ b ] + e 2 i φ { 8 i J 2 U ( 2 i Δ b + κ b ) [ 2 i ( U + Δ b ) + κ b ] [ 2 i ( Δ a + Δ b ) + κ a + κ b ] } .
η 1 Δ a 2 + η 2 Δ a + η 3 = 0 ,
Δ a , o p t ( 1 ) , ( 2 ) = η 2 ± η 2 2 4 η 1 η 3 2 η 1 .
U a , o p t = { 8 J 2 r 2 ( Δ a + Δ b ) + 2 cos ( 2 φ ) [ 4 Δ b 2 ( Δ a + Δ b ) + 2 Δ b κ a κ b + ( Δ a + 3 Δ b ) κ b 2 ] 4 J r cos ( φ ) [ 4 Δ b ( Δ a + Δ b ) + κ b ( κ a + κ b ) ] + 8 J r sin ( φ ) [ Δ a κ b + Δ b ( κ a + 2 κ b ) ] + sin ( 2 φ ) [ 8 Δ a Δ b κ b + κ b 2 ( κ a + κ b ) 4 Δ b 2 ( κ a + 3 κ b ) ] } / { 8 J 2 r 2 16 J r cos ( φ ) ( Δ a + Δ b ) 8 J r sin ( φ ) ( κ a + κ b ) + cos ( 2 φ ) [ 8 J 2 + 8 Δ b ( Δ a + Δ b ) 2 κ b ( κ a + κ b ) ] 4 sin ( 2 φ ) [ Δ a κ b + Δ b ( κ a + 2 κ b ) ] } ,
Δ b 4 + b 1 Δ b 3 + c 1 Δ b 2 + d 1 Δ b + e 1 = 0 ,
δ = J 4 ( r 2 1 ) 2 [ N 2 J 8 + N 1 + 16 J 2 κ 6 ( 1 + 3 r 2 ) ] 4096 , h = 1 12 [ 3 ( r 2 1 ) J 2 κ 2 ] , η = J 2 16 ( r 2 1 ) [ 9 ( r 2 1 ) J 2 8 κ 2 ] ,
A ¯ 10 = 2 e i φ b F a [ 2 r J + i e i φ ( κ + 2 i Δ ) ] 4 J 2 + ( κ + 2 i Δ ) 2 , A ¯ 20 = 2 2 e 2 i φ b F a 2 ( 4 i r J w 1 e i φ + w 2 + e 2 i φ w 3 ) w 4 [ ( κ + 2 i Δ ) 2 w 5 + 4 J 2 κ + 4 J 2 ( U + 2 Δ ) i ] , A ¯ 11 = 4 e 2 i φ b F a 2 ( 2 J e i φ + i r κ 2 r Δ ) ( i w 1 e i φ + w 2 2 r J ) w 4 [ ( κ + 2 i Δ ) 2 w 5 + 4 J 2 κ + 4 J 2 ( U + 2 Δ ) i ] ,
g a ( 2 ) ( τ ) = Tr S [ A ^ A ^ ρ ˜ ( τ ) ] ,
ρ ˜ ( τ ) = Tr E [ U T ( τ ) ρ ˜ ( 0 ) ρ E U T ( τ ) ] ,
| Ψ ˜ ( 0 ) = a ^ | Ψ ¯ ,
g a ( 2 ) ( τ ) = | A 10 ( τ ) | 2 | A ¯ 10 | 4 .
| Ψ ˜ ( 0 ) = A ¯ 10 | 00 + 2 A ¯ 20 | 10 + A ¯ 11 | 01 .
A 10 ( τ ) + A 01 ( τ ) = x ( τ ) , A 10 ( τ ) A 01 ( τ ) = y ( τ ) ,
x τ + u 1 x ( τ ) = λ 1 , y τ + u 2 y ( τ ) = λ 2 ,
x ( τ ) = x ( 0 ) e u 1 τ + λ 1 u 1 ( 1 e u 1 τ ) , y ( τ ) = y ( 0 ) e u 2 τ + λ 2 u 2 ( 1 e u 2 τ ) .

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